Media buyers are all thugs, right? East End bruisers who shout very loudly down the phone.

Well yes, there are some for whom this approach works very successfully. Not here though. I'll get personal now: the nicest thing a media sales person ever said to me was "the trouble with you, Vincent, is that you've always got a special reason why your client should get a better rate".  Good buying comes from good planning - buying the right media to reach the right people at the right time.  And yes, 80% and upwards of most advertising budgets goes into the media, so buying at the right price makes a crucial difference.
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